Let y be defined implicitly by the equation. -10x^(3)-8y^(3)-x-10 y=-91". " Use implicit differentiation to find (dy)/(dx).

Write Equation: Write down the given equation. The given equation is -10x^3 - 8y^3 - x - 10y = -91.Apply Differentiation: Apply implicit differentiation to both sides of the equation with respect to x. Differentiating term by term, we get: d/dx(-10x^3) + d/dx(-8y^3) + d/dx(-x) + d/dx(-10y) = d/dx(-91)Differentiate with Respect: Differentiate each term with respect to x. For the x terms, we use the power rule and the constant rule. For the y terms, we use the chain rule since y is a function of x. This gives us: -30x^2 - 24y^2(dy/dx) - 1 - 10(dy/dx) = 0Isolate Terms: Isolate terms with (dy/dx) on one side and the rest on the other side. -24y^2(dy/dx) - 10(dy/dx) = 30x^2 + 1Factor Out: Factor out (dy/dx) from the left side of the equation. (dy/dx)(-24y^2 - 10) = 30x^2 + 1Solve for dy/dx: Solve for (dy/dx). (dy/dx) = (30x^2 + 1) / (-24y^2 - 10)Check for Errors: Check for any mathematical errors in the differentiation and algebraic manipulation. No errors are found in the differentiation or algebraic manipulation.

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Let y be defined implicitly by the equation.
-10x^(3)-8y^(3)-x-10 y=-91". "
Use implicit differentiation to find
(dy)/(dx).

Let y be defined implicitly by the equation. $-10 x^{3}-8 y^{3}-x-10 y=-91 \text {. }$ $\newline$ Use implicit differentiation to find $\frac{d y}{d x}$ .

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Q. Let y be defined implicitly by the equation.

-10 x^{3}-8 y^{3}-x-10 y=-91 \text {. }

\newline

Use implicit differentiation to find

\frac{d y}{d x}

Write Equation: Write down the given equation. $\newline$ The given equation is $-10x^3 - 8y^3 - x - 10y = -91$ .
Apply Differentiation: Apply implicit differentiation to both sides of the equation with respect to $x$ . Differentiating term by term, we get: $\frac{d}{dx}(-10x^3) + \frac{d}{dx}(-8y^3) + \frac{d}{dx}(-x) + \frac{d}{dx}(-10y) = \frac{d}{dx}(-91)$
Differentiate with Respect: Differentiate each term with respect to $x$ . For the $x$ terms, we use the power rule and the constant rule. For the $y$ terms, we use the chain rule since $y$ is a function of $x$ . This gives us: $-30x^2 - 24y^2\frac{dy}{dx} - 1 - 10\frac{dy}{dx} = 0$
Isolate Terms: Isolate terms with $\frac{dy}{dx}$ on one side and the rest on the other side. $\newline$ $-24y^2\frac{dy}{dx} - 10\frac{dy}{dx} = 30x^2 + 1$
Factor Out: Factor out $(\frac{dy}{dx})$ from the left side of the equation. $\newline$ $(\frac{dy}{dx})(-24y^2 - 10) = 30x^2 + 1$
Solve for $\frac{dy}{dx}$ : Solve for $\frac{dy}{dx}$ . $\frac{dy}{dx} = \frac{30x^2 + 1}{-24y^2 - 10}$
Check for Errors: Check for any mathematical errors in the differentiation and algebraic manipulation. $\newline$ No errors are found in the differentiation or algebraic manipulation.